The Effect of Longitudinal Pre-Stretch and Radial Constraint on the Stress Distribution in the Vessel Wall: A New Hypothesis

It is well known that blood vessels shorten axially when excised. This is due to the perivascular tethering constraint by side branches and the existence of pre-stretch of blood vessels at the \textit {in situ} state. Furthermore, vessels are radially constrained to various extents by the surrounding tissues at physiological loading. Our hypothesis is that the axial pre-stretch and radial constraint by the surrounding tissue homogenizes the stress and strain distributions in the vessel wall. A finite element analysis of porcine coronary artery and rabbit thoracic aorta based on measured material properties, geometry, residual strain and physiological loading is used to compute the intramural stresses and strains. We systematically examined the effect of pre-stretch and external radial constraint in both vessels. Our results show that both stretching in the axial direction and compression in the radial direction lead to a more homogeneous strain and stress state in the blood vessel wall. A ``uniform biaxial strain' hypothesis is proposed for the blood vessel wall and the ramifications are discussed.     

Simulation of the behaviour of the L1 vertebra for different material properties and loading conditions

Three-dimensional finite element models of the thoracolumbar junction (T12–L2) and isolated L1 vertebra were developed to investigate the role of material properties and loading conditions on vertebral stresses and strains to predict fracture risk. The geometry of the vertebrae was obtained from computed tomography images. The isolated vertebra model included an L1 vertebra loaded through polymethylmethacrylate plates located at the top and bottom of the vertebra, and the segment model included T12 to L2 vertebrae and seven ligaments, fibrous intervertebral discs and facet joints. Each model was examined with both homogeneous and spatially varying bone tissue properties. Stresses and strains were compared for uniform compression and flexion. Including material heterogeneity remarkably reduced the stiffness of the isolated L1 vertebra and increased the magnitudes of the minimum principal strains and stresses in the mid-transverse section. The stress and strain distributions further changed when physiological loading was applied to the L1 vertebra. In the segment models, including heterogeneous material properties increased the magnitude of the minimum principal strain by 158% in the centre of the mid-transverse section. Overall, the inclusion of heterogeneity and physiological loading increased the magnitude of the strains up to 346% in flexion and 273% in compression.     

Boundary layer considerations in a multi-layer model for LDL accumulation

Abstract

Boundary layer effects for Low-Density Lipoprotein (LDL) concentration problems in a multi-layer artery model are analyzed in this work. Both a straight artery and aorta-iliac bifurcation are analyzed. Mass, momentum and species governing equations are based on the porous media theory and solved with the commercial finite-element based code COMSOL Multiphysics. For the straight artery, various inlet velocities, arterial sizes and intramural pressure values are investigated. Results are presented in terms of concentration profiles close to the lumen/endothelium interface and boundary layer thickness. It is shown that the boundary layer is affected by all of the three analyzed parameters. The results in this work will further clarify the concentration polarization effects imposed by the arterial wall.     

Concentration polarization of atherogenic lipids in the arterial system

Nomenclature c, Normalized LDL concentration (C*/C0); C0, incoming (bulk) LDL concentration (gr/cm3); Cw, LDL concentration on the luminal surface (gr/cm3); ,wC time average value of LDL concentration on the luminal surface (gr/cm3); D, diffusion coef-ficient of LDL (cm2/s); Q, blood flow rate (mL/s); 0R, average internal radius of the artery (cm); Re, Reynolds number (002/Run); Sc, Schmidt number (/Dn); t, normalized time (00*/tuR); u, normalized axial velocity (0*/uu); 0u, time a…     

Concentration polarization of atherogenic lipids in the arterial system

The transport of atherogenic lipids (LDL) in a straight segment of an artery with a semi-permeable wall was simulated numerically. The numerical analysis predicted that a mass transport phenomenon called ’concentration polarization’ of LDL might occur in the arterial system. Under normal physiological flow conditions, the luminal surface LDL concentration was 5%–14% greater than the bulk concentration in a straight segment of an artery. The luminal surface LDL concentration at the arterial wall was flow-dependent, varying linearly with the filtration rate across the arterial wall and inversely with wall shear rate. At low wall shear rate, the luminal surface LDL concentration was very sensitive to changes in flow conditions, decreasing sharply as wall shear rate increased. In order to verify the numerical analysis, the luminal surface concentration of bovine serum albumin (as a tracer macromolecule) in the canine carotid artery was measured in vitro by directly taking liquid samples from the luminal surface of the artery. The experimental result was in very good agreement with the numerical analysis. The authors believe that the mass transport phenomenon of ‘concentration polarization’ may indeed exist in the human circulation and play an important role in the localization of atherosclerosis.     

On the sensitivity of wall stresses in diseased arteries to variable material properties

Accurate estimates of stress in an atherosclerotic lesion require knowledge of the material properties of its components (e.g., normal wall, fibrous plaque, calcified regions, lipid pools) that can only be approximated. This leads to considerable uncertainty in these computational predictions. A study was conducted to test the sensitivity of predicted levels of stress and strain to the parameter values of plaque used in finite element analysis. Results show that the stresses within the arterial wall, fibrous plaque, calcified plaque, and lipid pool have low sensitivities for variation in the elastic modulus. Even a +/- 50% variation in elastic modulus leads to less than a 10% change in stress at the site of rupture. Sensitivity to variations in elastic modulus is comparable between isotropic nonlinear, isotropic nonlinear with residual strains, and transversely isotropic linear models. Therefore, stress analysis may be used with confidence that uncertainty in the material properties generates relatively small errors in the prediction of wall stresses. Either isotropic nonlinear or anisotropic linear models provide useful estimates, however the predictions in regions of stress concentration (e.g., the site of rupture) are somewhat more sensitive to the specific model used, increasing by up to 30% from the isotropic nonlinear to orthotropic model in the present example. Changes resulting from the introduction of residual stresses are much smaller.     

A study on large radial motion of arteries in vivo

This study analyses the radial periodic motion of an artery which is modelled as a thin cylinder of uniform cross-section subjected to dynamic inner pressure using the theory of finite deformation of elastic materials. The arterial tissue properties (anisotropy, homogeneity and incompressibility) are taken into account in an analysis based on the use of the strain energy function. The validity of the mathematical analysis is illustrated through numerical computation applying the available in vivo data for elastic constants of the canine middle descending thoracic aorta to the expressions for the intramural pressure and circumferential stresses obtained by solving the necessary equation of motion together with the boundary conditions. Results obtained in this study indicate very low stresses which suggest that the arteriosclerosis resulting from high stress gradients is effectively ruled out in this model.     

Time course of flow-induced adaptation of carotid artery biomechanical properties, structure and zero-stress state in the arteriovenous shunt

Numerous studies have provided evidence of diameter adaptation secondary to flow-overload, but with ambiguous findings vis à vis other morphological parameters and information on the biomechanical aspects of arterial adaptation is rather incomplete. We examined the time course of large-artery biomechanical adaptation elicited by long-term flow-overload in a porcine shunt model between the carotid artery and ipsilateral jugular vein. Post-shunting, the proximal artery flow was doubled and retained so until euthanasia (up to three months post-operatively), without pressure change. This hemodynamic stimulus induced lumen diameter enlargement, accommodated by elastin fragmentation and connective tissue accumulation, as witnessed by optical and confocal microscopy. Heterogeneous mass growth of the adventitia was observed at the expense of the media, associated with declining residual strains and opening angle at three months. The in vitro elastic properties of shunted arteries determined by inflation/extension testing were also modified, with the thickness-pressure curves shifted to larger thicknesses and the diameter-pressure curves shifted to larger diameters at physiologic pressures, resulting in normalization of intramural and shear stresses within fifteen and thirty days, respectively. We infer that the biomechanical adaptation in moderate flow-overload leads to normalization of intimal shear, without, however, restoring compliance and distensibility at mean in vivo pressure to control levels.     

Dynamic response of wall shear stress on the stenosed artery

The present study deals with an appropriate mathematical model of an artery in the presence of constriction in which the generated wall shear stress due to blood flow is analysed. The geometry of the stenosed arterial segment in the diseased state, causing malfunction of the cardiovascular system, is formed mathematically. The flowing blood contained in the stenosed artery is treated as non-Newtonian and the flow is considered to be two-dimensional. The motion of the arterial wall and its effect on local fluid mechanics is not ruled out from the present pursuit. The flow analysis applies the time-dependent, two-dimensional incompressible nonlinear Navier–Stokes equations for non-Newtonian fluids. The flow-field can be obtained primarily following the radial coordinate transformation, using the appropriate boundary conditions and finally adopting a suitable finite difference scheme numerically. The influences of flow unsteadiness, the arterial wall distensibility and the presence of stenosis on the flow-field and the wall shear stresses are quantified in order to indicate the susceptibility to atherosclerotic lesions and thereby to validate the applicability of the present theoretical model.     

Initial stress and nonlinear material behavior in patient-specific AAA wall stress analysis

Rupture risk estimation of abdominal aortic aneurysms (AAA) is currently based on the maximum diameter of the AAA. A more critical approach is based on AAA wall stress analysis. For that, in most cases, the AAA geometry is obtained from CT-data and treated as a stress free geometry. However, during CT imaging, the AAA is subjected to a time-averaged blood pressure and is therefore not stress free. The aim of this study is to evaluate the effect of neglecting these initial stresses (IS) on the patient-specific AAA wall stress as computed by finite element analysis. Additionally, the contribution of the nonlinear material behavior of the AAA wall is evaluated.Thirty patients with maximum AAA diameters below the current surgery criterion were scanned with contrast-enhanced CT and the AAA's were segmented from the image data. The mean arterial blood pressure (MAP) was measured immediately after the CT-scan and used to compute the IS corresponding with the CT geometry and MAP. Comparisons were made between wall stress obtained with and without IS and with linear and nonlinear material properties.On average, AAA wall stresses as computed with IS were higher than without IS. This was also the case for the stresses computed with the nonlinear material model compared to the linear material model. However, omitting initial stress and material nonlinearity in AAA wall stress computations leads to different effects in the resulting wall stress for each AAA. Therefore, provided that other assumptions made are not predominant, IS cannot be discarded and a nonlinear material model should be used in future patient-specific AAA wall stress analyses.     

Stress and strain adaptation in load-dependent remodeling of the embryonic left ventricle

Altered pressure in the developing left ventricle (LV) results in altered morphology and tissue material properties. Mechanical stress and strain may play a role in the regulating process. This study showed that confocal microscopy, three-dimensional reconstruction, and finite element analysis can provide a detailed model of stress and strain in the trabeculated embryonic heart. The method was used to test the hypothesis that end-diastolic strains are normalized after altered loading of the LV during the stages of trabecular compaction and chamber formation. Stage-29 chick LVs subjected to pressure overload and underload at stage 21 were reconstructed with full trabecular morphology from confocal images and analyzed with finite element techniques. Measured material properties and intraventricular pressures were specified in the models. The results show volume-weighted end-diastolic von Mises stress and strain averaging 50–82 % higher in the trabecular tissue than in the compact wall. The volume-weighted-average stresses for the entire LV were 115, 64, and 147 Pa in control, underloaded, and overloaded models, while strains were 11, 7, and 4 %; thus, neither was normalized in a volume-weighted sense. Localized epicardial strains at mid-longitudinal level were similar among the three groups and to strains measured from high-resolution ultrasound images. Sensitivity analysis showed changes in material properties are more significant than changes in geometry in the overloaded strain adaptation, although resulting stress was similar in both types of adaptation. These results emphasize the importance of appropriate metrics and the role of trabecular tissue in evaluating the evolution of stress and strain in relation to pressure-induced adaptation.     

A model of stress-induced geometrical remodeling of vessel segments adjacent to stents and artery/graft anastomoses

The mismatch between the elastic properties and initial geometry of a host artery and an implanted stent or graft cause significant stress concentration at the zones close to junctions. This may contribute to the often observed intimal hyperplasia, resulting in late lumen loss and eventual restenosis. This study proposes a mathematical model for stress-induced thickening of the arterial wall at the zones close to an implanted stent or graft. The host artery was considered initially as a cylindrical shell with constant thickness that was clamped to the stent or graft, which was assumed to be non-deformable in the circumferential direction. It was assumed that the abnormal circumferential and axial stresses due to the bending of the arterial wall cause wall thickening that tends to restore the stress state close to that existing far from the junction. The linear equations of a cylindrical shell with variable thickness were coupled to an evolution equation for the wall thickness. These equations were solved numerically and a parametric study was performed using finite difference method and explicit time step. The results show that the remodeling process is self-limiting and leads to local thickening that gradually decreases with distance from the edge of the stent/graft. Model predictions were tested against morphological findings existing in the literature. Recommendations on stent designs that reduce stress concentrations are discussed.     

Effect of surrounding tissue on vessel fluid and solid mechanics

There is no doubt that atherosclerosis is one of the most important health problems in the Western Societies. It is well accepted that atherosclerosis is associated with abnormal stress and strain conditions. A compelling observation is that the epicardial arteries develop atherosclerosis while the intramural arteries do not. Atherosclerotic changes involving the epicardial portion of the coronary artery stop where the artery penetrates the myocardium. The objective of the present study is to understand the fluid and solid mechanical differences between the two types of vessels. A finite element analysis was employed to investigate the effect of external tissue contraction on the characteristics of pulsatile blood flow and the vessel wall stress distribution. The sequential coupling of fluid-solid interaction (FSI) revealed that the changes of flow velocity and wall shear stress, in response to cyclical external loading, appear less important than the circumferential stress and strain reduction in the vessel wall under the proposed boundary conditions. These results have important implications since high stresses and strains can induce growth, remodeling, and atherosclerosis; and hence we speculate that a reduction of stress and strain may be atheroprotective. The importance of FSI in deformable vessels with pulsatile flow is discussed and the fluid and solid mechanics differences between epicardial and intramural vessels are highlighted.     

Effect of aging on mechanical stresses,deformations, and hemodynamics in human femoropopliteal artery due to limb flexion

Femoropopliteal artery (FPA) reconstructions are notorious for poor clinical outcomes. Mechanical and flow conditions that occur in the FPA with limb flexion are thought to play a significant role, but are poorly characterized. FPA deformations due to acute limb flexion were quantified using a human cadaver model and used to build a finite element model that simulated surrounding tissue forces associated with limb flexion-induced deformations. Strains and intramural principal mechanical stresses were determined for seven age groups. Computational fluid dynamics analysis was performed to assess hemodynamic variables. FPA shape, stresses, and hemodynamics significantly changed with age. Younger arteries assumed straighter positions in the flexed limb with less pronounced bends and more uniform stress distribution along the length of the artery. Even in the flexed limb posture, FPAs younger than 50 years of age experienced tension, while older FPAs experienced compression. Aging resulted in localization of principal mechanical stresses to the adductor hiatus and popliteal artery below the knee that are typically prone to developing vascular pathology. Maximum principal stresses in these areas increased threefold to fivefold with age with largest increase observed at the adductor hiatus. Atheroprotective wall shear stress reduced after 35 years of age, and atheroprone and oscillatory shear stresses increased after the age of 50. These data can help better understand FPA pathophysiology and can inform the design of targeted materials and devices for peripheral arterial disease treatments.     

An argument for the use of multiple segment stents in curved arteries

Stenting of curved arteries is generally perceived to be more challenging than straight vessels. Conceptually implanting multiple shorter stents rather than a single longer stent into such a curved artery represents a promising concept, but little is known about the impact of such an approach. The objective of this study is to evaluate the effectiveness of using a multiple segment stent rather than a single long stent to dilate a curved artery using the finite element method. A double segment stent (DSS) and a single segment stent (SSS) were modeled. The stents were compared when expanded into a model of a curved artery. The model predicts that the DSS provides higher flexibility, more conformity, and lower recoil in comparison to the SSS. The volume of arterial tissue experiencing high levels of stress due to stent implantation is also reduced for the DSS. It is suggested that a multiple segment stenting system is a potential solution to the problem of higher rates of in-stent restenosis in curved arteries and mechanically challenging environments.     

The choice of a constitutive formulation for modeling limb flexion-induced deformations and stresses in the human femoropopliteal arteries of different ages

Open and endovascular treatments for peripheral arterial disease are notorious for high failure rates. Severe mechanical deformations experienced by the femoropopliteal artery (FPA) during limb flexion and interactions between the artery and repair materials play important roles and may contribute to poor clinical outcomes. Computational modeling can help optimize FPA repair, but these simulations heavily depend on the choice of constitutive model describing the arterial behavior. In this study finite element model of the FPA in the standing (straight) and gardening (acutely bent) postures was built using computed tomography data, longitudinal pre-stretch and biaxially determined mechanical properties. Springs and dashpots were used to represent surrounding tissue forces associated with limb flexion-induced deformations. These forces were then used with age-specific longitudinal pre-stretch and mechanical properties to obtain deformed FPA configurations for seven age groups. Four commonly used invariant-based constitutive models were compared to determine the accuracy of capturing deformations and stresses in each age group. The four-fiber FPA model most accurately portrayed arterial behavior in all ages, but in subjects younger than 40 years, the performance of all constitutive formulations was similar. In older subjects, Demiray (Delfino) and classic two-fiber Holzapfel–Gasser–Ogden formulations were better than the Neo-Hookean model for predicting deformations due to limb flexion, but both significantly overestimated principal stresses compared to the FPA or Neo-Hookean models.     

Velocity and wall shear stress patterns in the human right coronary artery.

Blood flow dynamics in the human right coronary artery have not been adequately quantified despite the clinical significance of coronary atherosclerosis. In this study, a technique was developed to construct a rigid flow model from a cast of a human right coronary artery. A laser photochromic method was used to characterize the velocity and wall shear stress patterns. The flow conditions include steady flow at Reynolds numbers of 500 and 1000 as well as unsteady flow with Womersley parameter and peak Reynolds number of 1.82 and 750, respectively. Characterization of the three-dimensional geometry of the artery revealed that the largest spatial variation in curvature occurred within the almost branch-free proximal region, with the greatest curvature existing along the acute margin of the heart. In the proximal segment, high shear stresses were observed on the outer wall and lower, but not negative, stresses along the inner wall. Low shear stress on the inner wall may be related to the preferential localization of atherosclerosis in the proximal segment of the right coronary artery. However, it is possible that the large difference between the outer and inner wall shear stresses may also be involved.     

Pulsatile arterial wall-blood flow interaction with wall pre-stress computed using an inverse algorithm

Background

The computation of arterial wall deformation and stresses under physiologic conditions requires a coupled compliant arterial wall-blood flow interaction model. The in-vivo arterial wall motion is constrained by tethering from the surrounding tissues. This tethering, together with the average in-vivo pressure, results in wall pre-stress. For an accurate simulation of the physiologic conditions, it is important to incorporate the wall pre-stress in the computational model. The computation of wall pre-stress is complex, as the un-loaded and un-tethered arterial shape with residual stress is unknown. In this study, the arterial wall deformation and stresses in a canine femoral artery under pulsatile pressure was computed after incorporating the wall pre-stresses. A nonlinear least square optimization based inverse algorithm was developed to compute the in-vivo wall pre-stress.

Methods

First, the proposed inverse algorithm was used to obtain the un-loaded and un-tethered arterial geometry from the unstressed in-vivo geometry. Then, the un-loaded, and un-tethered arterial geometry was pre-stressed by applying a mean in-vivo pressure of 104.5 mmHg and an axial stretch of 48% from the un-tethered length. Finally, the physiologic pressure pulse was applied at the inlet and the outlet of the pre-stressed configuration to calculate the in-vivo deformation and stresses. The wall material properties were modeled with an incompressible, Mooney-Rivlin model derived from previously published experimental stress-strain data (Attinger et al., 1968).

Results

The un-loaded and un-tethered artery geometry computed by the inverse algorithm had a length, inner diameter and thickness of 35.14 mm, 3.10 mm and 0.435 mm, respectively. The pre-stressed arterial wall geometry was obtained by applying the in-vivo axial-stretch and average in-vivo pressure to the un-loaded and un-tethered geometry. The length of the pre-stressed artery, 51.99 mm, was within 0.01 mm (0.019%) of the in-vivo length of 52.0 mm; the inner diameter of 3.603 mm was within 0.003 mm (0.08%) of the corresponding in-vivo diameter of 3.6 mm, and the thickness of 0.269 mm was within 0.0015 mm (0.55%) of the in-vivo thickness of 0.27 mm. Under physiologic pulsatile pressure applied to the pre-stressed artery, the time averaged longitudinal stress was found to be 42.5% higher than the circumferential stresses. The results of this study are similar to the results reported by Zhang et al., (2005) for the left anterior descending coronary artery.

Conclusions

An inverse method was adopted to compute physiologic pre-stress in the arterial wall before conducting pulsatile hemodynamic calculations. The wall stresses were higher in magnitude in the longitudinal direction, under physiologic pressure after incorporating the effect of in-vivo axial stretch and pressure loading.

     

A Novel Tram Stent Method in the Treatment of Coronary Bifurcation Lesions – Finite Element Study

A novel stent was designed for the treatment of coronary bifurcation lesion, and it was investigated for its performance by finite element analysis. This study was performed in search of a novel method of treatment of bifurcation lesion with provisional stenting. A bifurcation model was created with the proximal vessel of 3.2 mm diameter, and the distal vessel after the side branch (2.3 mm) was 2.7 mm. A novel stent was designed with connection links that had a profile of a tram. Laser cutting and shape setting of the stent was performed, and thereafter it was crimped and deployed over a balloon. The contact pressure, stresses on the arterial wall, stresses on the stent, the maximal principal log strain of the main artery and the side-branch were studied. The study was performed in Abaqus, Simulia. The stresses on the main branch and the distal branch were minimally increased after deployment of this novel stent. The side branch was preserved, and the stresses on the side branch were lesser; and at the confluence of bifurcation on either side of the side branch origin the von-Mises stress was marginally increased. The stresses and strain at the bifurcation were significantly lesser than the stresses and strain of the currently existing techniques used in the treatment of bifurcation lesions though the study was primarily focused only on the utility of the new technology. There is a potential for a novel Tram-stent method in the treatment of coronary bifurcation lesions.     

Effect of residual stress and heterogeneity on circumferential stress in the arterial wall

Quantifying the stress distribution through the arterial wall is essential to studies of arterial growth and disease. Previous studies have shown that both residual stress, as measured by opening angle, and differing material properties for the media-intima and the adventitial layers affect the transmural circumferential stress (sigma theta) distribution. Because a lack of comprehensive data on a single species and artery has led to combinations from multiple sources, this study determined the sensitivity of sigma theta to published variations in both opening angle and layer thickness data. We fit material properties to previously published experimental data for pressure-diameter relations and opening angles of rabbit carotid artery, and predicted sigma theta through the arterial wall at physiologic conditions. Using a one-layer model, the ratio of sigma theta at the internal wall to the mean sigma theta decreased from 2.34 to 0.98 as the opening angle increased from 60 to 130 deg. In a two-layer model using a 95 deg opening angle, mean sigma theta in the adventitia increased (112 percent for 25 percent adventitia) and mean sigma theta in the media decreased (47 percent for 25 percent adventitia). These results suggest that both residual stress and wall layers have important effects on transmural stress distribution. Thus, experimental measurements of loading curves, opening angles, and wall composition from the same species and artery are needed to accurately predict the transmural stress distribution in the arterial wall.